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 A141821 Least number k < n and coprime to n such that the largest term of the continued fraction of k/n is as small as possible. 5
 1, 2, 3, 2, 5, 5, 3, 7, 3, 8, 5, 5, 11, 4, 7, 12, 13, 7, 9, 8, 17, 7, 7, 7, 19, 19, 23, 12, 11, 12, 25, 10, 13, 27, 11, 10, 9, 14, 11, 29, 11, 31, 31, 19, 17, 34, 37, 18, 19, 40, 41, 14, 17, 21, 15, 16, 17, 18, 47, 17, 23, 46, 45, 46, 25, 49, 49, 50, 29, 26, 19, 27, 31, 29, 55, 34, 61 (list; graph; refs; listen; history; text; internal format)
 OFFSET 2,2 COMMENTS See A141822 for the value of the largest term in the continued fraction of a(n)/n. Zaremba conjectured that the largest value is 5. REFERENCES R. K. Guy, Unsolved problems in number theory, F20. S. K. Zaremba, ed., "Applications of number theory to numerical analysis," Proceedings of the Symposium at the Centre for Research in Mathematics, University of Montreal, Academic Press, New York, London (1972). LINKS Robin Visser, Table of n, a(n) for n = 2..10000 (terms n = 2..2000 from T. D. Noe). T. W. Cusick, Zaremba's conjecture and sums of the divisor function, Math. Comp. 61 (1993), 171-176. Takao Komatsu, On a Zaremba's conjecture for powers, Sarajevo J. Math. 1 (2005), 9-13. EXAMPLE For n=7, the six continued fractions for k/7 are (0, 7), (0, 3, 2), (0, 2, 3), (0, 1, 1, 3), (0, 1, 2, 2) and (0, 1, 6). It is easy to see that the fifth one, for 5/7, has the smallest maximum term, 2. Hence a(7)=5. MATHEMATICA Table[k=Select[Range[n-1], GCD[ #, n]==1&]; c=ContinuedFraction[k/n]; mx=Max/@c; mn=Min[mx]; k[[Position[mx, mn, 1, 1][[1, 1]]]], {n, 2, 100}] CROSSREFS Sequence in context: A357987 A135737 A125179 * A144308 A144307 A144310 Adjacent sequences: A141818 A141819 A141820 * A141822 A141823 A141824 KEYWORD nonn AUTHOR T. D. Noe, Jul 08 2008 STATUS approved

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Last modified March 5 03:04 EST 2024. Contains 370537 sequences. (Running on oeis4.)